Composite Supersymmetries in low-dimensional systems

Starting from a N=1 scalar supermultiplet in 2+1 dimensions, we demonstrate explicitly the appearance of induced N=1 vector and scalar supermultiplets of composite operators made out of the fundamental supersymmetric constituents. We discuss an extension to a N=2 superalgebra with central extension, due to the existence of topological currents in 2+1 dimensions. As a specific model we consider a supersymmetric $CP^1$ $\sigma$-model as the constituent theory, and discuss the relevance of these results for an effective description of the infrared dynamics of planar high-temperature superconducting condensed matter models with quasiparticle excitations near nodal points of their Fermi surface.


Introduction
Supersymmetry is a symmetry that links integer spin excitations (bosons) to half-integer ones (fermions), and, in certain circumstances, provides a satisfactory control of quantum fluctuations, to the extent that many exact analytic results can be obtained on the phase structure of certain relativistic field theories 1 . From this point of view, supersymmetry is expected mainly to appear, if at all, in theories of the fundamental interactions of Nature, which are by construction relativistic. In fact supersymmetry was a helpful tool for understanding the non-perturbative structure of strongly coupled non-abelian gauge theories, of relevance to the weak and strong interactions. Seiberg and Witten [1] have managed very effectively to exploit extended N = 2 four-dimensional supersymmetry in SU(2) gauge theory so as to obtain complete non-perturbative information on the phasestructure of the theory. This work opened the way to more exact results, some of which pertained to theories in space-time dimensions different from four. There has been considerable interest, for instance, in N = 2 supersymmetric gauge theories in three-dimensional space-times [2], where some exact non-perturbative information can also be obtained. At present it is N = 1 supersymmetry, which seems to be of phenomenological relevance to Nature in four dimensions, and unfortunately for it there are no such exact results in any dimension. Nevertheless, it may be hoped that, by viewing N = 1 supersymmetry as a broken phase of an extended supersymmetry, some exact information on the phase structure can be extracted. Attempts in this direction have already been made in the literature, e.g. four-dimensional softly broken N = 2 QCD [3]. Given the necessity of relativistic systems, dynamical supersymmetry was thought to be irrelevant for theories of condensed matter. This is due to the fact that the majority of systems in condensed matter involve non-relativistic excitations around Fermi surfaces of finite-extent. For high-temperature superconductors, however, which are known to be d-wave superconductors (with nodes in the superconducting gap) it has been found that as the doping is lowered and the compound becomes non-superconducting, a pseudogap region is entered where there are nodes at the Fermi surface. This prompted the use of relativistic theories for describing the lowlying excitations near the nodes [4,5,6]. As a result, from the perspective of obtaining exact analytic information on the phase structure of such systems it is interesting to examine situations where the effective relativistic theories of such nodal excitations possess, in addition, dynamical supersymmetry.The hope is that such results may also be useful in yielding important information on excitations, even away from the nodes. Such attempts have been initiated in [7,8] in the context of the so-called spin-charge separation phase of planar antiferromagnets [9], where the fundamental degrees of freedom are not elelctrons, but rather objects carrying only spin-(spinon) or only electric-charge-(holon) degrees of freedom. In terms of such excitations the continuum low-energy effective field theory describing the dynamics of the antiferromagnets is a CP 1 σ-model coupled to Dirac fermions. The z-magnons of the σ-model represent spinons, while the Dirac fermions represent holon degrees of freedom, which carry electric charge only [5]. The presence of CP 1 field theory necessitates the appearance of non-propagating U S (1) gauge interactions, which from the effective field theory viewpoint are viewed as interactions with infinitely strong gauge coupling (due to the absence of gauge kinetic (i.e. Maxwell) terms). In some cases [5], the nodal liquid is described in a 'particle-hole' symmetric formalism which allows the existence of additional non-Abelian gauge interactions of SU(2) type, representing the redundancy inherent in the ansatz of the spinon and holon degrees of freedom. In [8] it has been asserted that the effective theory of holons can be represented by Dirac fermions near the nodes of a Fermi surface provided the SU(2) gauge interactions are weakly coupled. This will be assumed throughout this work.
The infinitely strong U S (1) interactions can be integrated out to give [5] effective (dual) theories of spinon and holons comprising of composite objects made out of these fundamental constituents. These represent the effective physical degrees of freedom of the nodal system. In [8] the conditions for dynamical supersymmetry between spinon and holon constituent degrees of freedom in such effective theories have been analysed. The important point to note is that specific points in the parameter space, depending on particular doping concentrations, allow for a scalar N = 1 three-dimensional supersymmetry between spinon (z-magnon) and holon (Ψ Dirac fermions) degrees of freedom. In this we study the induced composite supersymmetry in the effective theory. As we shall show, the existence of a scalar N = 1 supermultiplet for spinons and holons suffices to induce N = 1 scalar and vector supermultiplets at a composite level in the dual theory (obtained from integration of the U S (1) interactions). It should be stressed that here we do not supersymmetrize the U S (1) interactions. As we shall see, due to the three dimensionality of the theory, it is possible obtain centrally extended N = 2 supersymmetries at both the constituent and the composite level, with the central extension being provided by appropriate topological 'charges' to be defined below [10,11]. The N = 2 supersymmetry couples the two N = 1 multiplets of the composite supersymmetry via gauge interactions, leading to interesting models. One possible model is the abelian Higgs model, while under different circumstances, to be discussed in detail later, a broken phase of a non-abelian SU(2) Georgi-Glashow model obtains. These two models can lead to different physics.
The structure of this article is as follows: in the next section we shall discuss notations and conventions pertaining to a scalar N = 1 supersymmetry at a constituent spinon-holon level. In section 3 we shall review the appearance of composite operators in these systems, discussed in [5]. In section 4 we shall study an N = 1 supersymmetry at a composite level, induced by the N = 1 scalar supersymmetry at a constituent spinon-holon level [7,8], and in particular we shall define the associated scalar supermultiplet (containing the scalar composites of [5]). In section 5 we shall discuss the induced N = 1 vector composite supermulitplet containing the vector composites of [5]. In section 6 we shall discuss the elevation of these supersymmetries in three dimensions into centrally extended ones with the central extension being identified with the topological charge [10,11]. In section 7 we shall argue that the appropriate effective composite theory is a N = 2 supersymmetric abelian-Higgs-like model, and discuss how this may lead to exact results on superconducting properties of the model at the supersymmetric points. The latter implies the existence of composite complex fermions, which are capable of ensuring the exact masslessness of an appropriate unbroken abelian subgroup of SU(2) which will be denoted by U 3 (1) and will be crucial for superconductivity by a mechanism promulgated before [4]. The superconducting point of the nodal liquid is argued to be a quantum critical point in the phase space of the model. A discussion on a pseudogap phase will also be made. Conclusions and outlook will be presented in section 8.
2 Scalar N = 1 Supersymmetry of spinon-holon constituents In [8] a detailed microscopic model was presented, and the associated continuum effective theory of doped antiferromagnets with spin-charge separation was derived. The constituent spinon and holon degrees of freedom formed a scalar N = 1 supermultiplet at certain points of the parameter space of the microscopic model, depending on the doping concentration.
In this section we set up the notation and conventions in 2 + 1 dimensions that will be used in this paper and also review the earlier work. The Dirac matrices are 2 × 2 and are given by where σ a are the Pauli matrices. Hence {γ µ , γ ν } = 2η µν with η µν = diag (−1, 1, 1) and µ, ν = 0, 1, 2. We raise and lower Dirac indices with the real antisymmetric tensor whose components are given by ǫ αβ = (−γ 0 ) α β = ǫ αβ . This convention has the property that real spinors remain real under the operation of raising and lowering indices.
It is important to notice that in the model of [8], as a result of the antiferromagnetic structure, there are two complex supermultiplets 2 . Hence starting with 4 N = 1 real scalar supermultiplets, we can construct 2 N = 1 complex scalar superfields Z a , a = 1, 2, made out of complex supermultiplets (z a , f a , ψ a ) which transform as For a product of complex spinors it is useful to note that and similarly that We also have the Fierz identity which is valid for any set of spinors χ i , i = 1, 2, 3, 4. We remind the important property concerning the antisymmetric tensor ǫ µνρ : Finally, if we write explicitely the 2 × 2 matrix χ 1 χ 2 , for any spinors χ 1 and χ 2 , we find the following relations: where 1 is the 2 × 2 unit matrix in Dirac space. We thus find that for any spinors χ 1 and χ 2 The associated long wavelength dynamics is represented by a supersymmetric σ-model which is described by the superfield lagrangian: where D α = ∂ ∂θ α + θ β / ∂ αβ denotes the supercovariant derivative. The superfield Z satisfies the constraint Z † a Z a = 1. In terms of components the constraint becomes:

Composite Meson Fields
In the path integral the integration of the strongly-coupled U S (1) gauge interactions lead to the appearance of composite operators [5], of 'meson' and 'baryon' type, comprising of holon constituents. In what follows we shall concentrate exclusively on meson type operators [5]: if we organize the fermions (holons) into a 'colour' doublet χ = (ψ 1 , ψ 2 ), built out of the two (two-component) fermion species ψ 1 and ψ 2 (corresponding to the two antiferromagnetic sublattices), then the bilinears (a = 1, 2, 3) transform as triplets under SU (2), where Σ a = σ a ⊗ 1 and Σ a µ = σ a ⊗ γ µ . On the other hand, the SU(2) singlets are given by the bilinears: where Γ µ = γ µ ⊗ 1. Note that S is a parity violating mass term, and S µ a four-component fermion-number current. On the other hand, the parity-invariant fermion mass term transforms as a triplet under the group SU(2). In models with a gauged SU(2) symmetry among spinon and holons, as required by a particle-hole symmetric spin-charge separation ansatz [5], the dynamical generation of a parity-invariant holon mass-gap will induce a dynamical breaking of the SU(2) gauge symmetry down to a U(1) compact subgroup. On ignoring non-perturbative effects, such a phase would be superconducting by the anomaly mechanism of [4]. However, due to the compact nature of the U(1) ⊂ SU(2) subgroup that is left unbroken, there are monopole-instanton effects that in general may be responsible for a small but non-zero mass of the U(1) gauge boson. This will spoil superconductivity, thereby leading to a pseudogap phase for the nodal liquid, as discussed in detail in [5]. Superconductivity, on the other hand, requires an exactly massless U(1) photon. Such a case arises in the Georgi-Glashow supersymmetric model of [13]. In [5] we did not discuss spinon contributions to the composite operators, because we worked in a phase where the spinon gap was much larger than the fermion (holon) dynamically generated mass gap. In the supersymmetric situation [8], any mass gaps of spinons and holons are of equal magnitude. It is natural to consider the effect of spinon composites could affect the mesons (10,11). This can be answered by invoking supersymmetry at a composite level. Since the mesons are obtained in [5] by integrating out a non-dynamical gauge group, the natural thing to assume is that the supersymmetry at a constituent magnon-holon level will be preserved at a composite level, and this will define the appropriate spinon contributions to the composites. In what follows we shall concentrate on the SU(2) triplet composites, which are the ones that could possibly couple to the gauge SU(2) interactions present in the particle-hole symmetric formalism of spin-charge separation [5]. As we shall demonstrate below, the bilinear composites can lead only to two decoupled N = 1 supermultiplets at a composite level, a scalar and a vector. Coupling of these two supermultiplets can only be seen if higher order constituent operators appear in the definition of the composite operators (10) and will be discussed in section 7. Such a coupling will allow the appearance of a N = 2 dynamical supersymmetry, with a central charge that coincides with the topolgical charge [10,11] defined by the vector fields in the three-dimensional case at hand.

N=1 composite scalar supermultiplet
Let us now consider a scalar composite Φ = ψ 1 ψ 2 . Φ defined in this way is complex but has the generic composite structure that we will study; the real scalar composites will be considered at the end of this section. The supersymmetric transform of Φ induced by the spinons and holons is δΦ = εΨ where The supersymmetric transform of Ψ is δΨ =F ε + M (+) ε wherẽ Using Eq. (7), we can also write such that δΨ = F ε + / ∂Φε where The supersymmetric transform of the auxilliary field F is which closes the superalgebra, since we have found the composite superpartners F and Ψ of the field Φ such that Such a closure of the scalar superalgebra is also valid for the SU(2) real triplet (a = 1, 2, 3) and the SU(2) real singlet with the notations introduced in the previous section.

N=1 composite vector supermultiplet
The supersymetric transformations for a N = 1 Abelian vector supermultiplet are in the Wess-Zumino gauge where λ is a real gaugino and f µν = ∂ [µ a ν] . Let us consider the composite vector field and its Abelian field strength F µν . The vector field (21) is complex but has the basic composite structure from which we will construct the real gauge fields at the end of this section. We will look for the supersymmetric transform of A µ , but before that we will derive the following property of F µν (see Eq.(24)): using the Fiertz identity (4), we can see that for any real Grassmann variables θ 1 , θ 2 we have and thus We The supersymmetric transform of A µ is now where the gaugino Λ is given by and has the following supersymmetric transform with Then, from the first of Eqs. (6), we can also write that for any spinor χ 1 and χ 2 such that and we finally find with Eqs. (24) and (27) Now let us come back to the transformation (25). The latter is the expected one up to an Abelian gauge transformation (total derivative): if we define the scalar ρ such that its supersymmetric transform is δρ = z ⋆ 1 εψ 2 − z 2 εψ ⋆ 1 , then we can write (25) as such that together with (31), we have defined a N = 1 vector composite supermultiplet, up to a gauge transformation. This result can be applied to the SU(2) real triplet (a = 1, 2, 3) and the SU(2) real singlet where φ = (z 1 , z 2 ) is a two-component scalar field and the other notations were introduced in section 3. We remind that the field strength F a µν appearing then in Eq.(31) is the Abelian one for each a (i.e. we obtain three independent Abelian composite supermultiplets).

Current supermultiplet and N=2 extended superalgebra
A specific feature of 2+1 dimensions is that we can always construct a topological conserved current J µ , starting from a vector A µ . This topological current is In general it can be shown [10] that the current J µ belongs to a new supermultiplet containing A µ and a spinorial currentS α µ . As a result, the N = 1 supersymmetry is centrally extended, with the central extension being provided by the topological charge associated with the current J µ [10]. Such a centrally extended supersymmetry characterizes the constituent spin-charge separating CP 1 supersymmetric model (8) of [8], according to the above mentioned general argument. Details on this can be found in the litterature [10] and will not be repeated here. We only mention that, in this case, the topological current belongs to a current supermultiplet J , which is defined in terms of the Z-scalar-superfields incorporating the magnon z and spinon ψ degrees of freedom: where α, β are spinorial indices, and D denotes the chiral superspace derivative. Due to the basic identity one can see that this current supermultiplet is identically conserved, i.e. D α J α = 0, without the use of the equations of motion. This framework can be applied at the composite level.
There is again a current supermultiplet involving the topological current constructed out of a composite vector field A a µ , a = 1, 2, 3 in (10). Although there are three such currents, our interest is in the effective dual theory of degrees of freedom which are massless at a perturbative level. As discussed in detail in [5] due to spontaneous symmetry-breaking, only one of the vector fields A a µ , a = 1, 2, 3 will remain massless in perturbation theory. From simulations this is also the case for the composite A µ 's [5]. Hence it is plausible to assume that the vector composite fields are gauged. In what follows we shall denote this vector field by the generic symbol A µ without any component index. Our analysis formally holds for any of the three components of the composite vector field (10). In particular, at a composite level we are interested in demonstrating that the current J µ = ǫ µνρ ∂ ν A ρ belongs to a new supermultiplet containing A µ and a spinorial composite currentS α µ . To show this let us return to the supersymmetric transform of A µ seen in the previous section and define the spinorial currentS α µ by such thatS α µ = γ µ Λ α . Since the translations commute with the supersymetric transformations,S µ is actually a conserved current in the specific gauge ∂ µ A µ = 0, in which by definition A µ is also conserved and we have / ∂Λ = 0. From the discussion in the previous section it folows that the supersymmetric transform ofS α µ is Let us now look at the supersymmetric transform of J µ .
Thus in the gauge ∂ µ A µ = 0, we have a conserved current supermultiplet satisfying the supersymmetric transformations Equations (41) form a closed superalgebra, since two successive supersymetric transformations applied on any of the currents lead to a translation of this current: It must be stressed that this result is independent of the model that we consider for the composite fields, since the supersymmetric transformations (1) concern the original microscopic fields z a and ψ a . Let us suppose that we have a Lagrangian for the composite vector field A µ . Then we can define the corresponding spinorial Noether's current S α µ and the associated supercharge Q α = d 2 xS α 0 . The new spinorial currentS α µ enables us to define a new superchargẽ Q α = d 2 xS α 0 . We will show that the anticommutator {Q α ,Q β } contains an antisymmetric part, proportionnal to the topological charge, which thus defines a N = 2 supersymmetric structure. From the infinitesimal transformation (39), we know that and thus Since the inverse of ǫ γα is ǫ αγ = (−γ 0 ) α γ , we have then The anticommutator between the two supergenerators is therefore where k = 1, 2 is a spatial index, and T = d 2 xJ 0 is the topological charge [10,11]. Since the matrices γ k are symmetric, (46) also implies which indicates a N = 2 superalgebra structure. However, it must be stressed that the mere appearance of a current supermultiplet does not necessarily imply a trully N = 2 extended dynamical supersymmetry in the physical spectrum of the model. This can only happen if the topological current is an independent quantity. In the case of the composite model discussed here, the topological current is constructed out of the vector composites, and hence, in order to promote the N = 2 structures to a true dynamical supersymmetry of the spectrum we must discuss in detail the dynamics of the composite model, in terms of the form of the associated Lagrangian. This will be the topic of the next section, where we shall see that the existence of a true N = 2 dynamical supersymmetry implies additional constraints for the coupling constants of the model [11].

N=2 Supersymmetric Composite Abelian Higgs model
As we have discussed in previous sections, the existence of constituent supersymmetries implies, via the appropriate transformation laws, the existence of fermionic composite excitations, which are parts of N = 1 supersymmetric composite multiplets of scalar and vector type. At a bilinear composite field level, to which we restricted our attention for the purposes of the current work, we are unable to couple these two types of supermultiplets. Such a coupling is provided by the gauge fields themselves. As we shall argue below, the latter can be identified with the vector composites. Indeed the viewing of the vector composites as gauge fields was essential for the closure of the N = 1 vector supermultiplets in section 5 since the closure of the vector supermultiplet under the constituent supersymmetry transformations occurs only up to abelian gauge transformations. Notice that the identification of the vector composites with SU(2) gauge fields leads to a dynamical breaking of the SU(2) gauge group to a compact abelian subgroup U 3 (1), in a way explained in detail in [5], and reviewed below. Such a gauge coupling would necessitate higher-thanbilinear constituent-field interactions among spinons and holons in the definition of the various composite fields (10), (11). At a composite level this would imply six order terms of constituent field operators in the constituent lagrangian, which are irrelevant operators at low energies (i.e. in the infrared), at least from a naive renormalization group point of view. One might then hope that, as far as the underlying constituent theory of holons and spinons is concerned, the infrared fixed point universality class will not be affected. For instructive purposes we first review briefly the non-Abelian ansatz for spin-charge separation of [5]. The particle-hole symmetric ansatz implies that the electron operators are composites of spinon and holon constituents in the form: which is valid at each lattice site of the underlying microscopic theory. Here c a , a = 1, 2 are electron operators, z a , a = 1, 2 are magnons (spinons), and χ a , a = 1, 2 are Grassmann numbers on the lattice representing electrically-charged holon excitations. The dynamical nodal theory depends only on the operators ξ [5], even away from the half-filling case, and thus there is a dynamical non-abelian gauge symmetry of SU(2) type, expressed by the invariance of the ansatz, and thus ξ, under simultaneous local SU(2) rotations of the spinon and holon parts: where h i ∈ SU(2) and i denotes lattice site.
In addition, there is a strongly-coupled dynamical U S (1) gauge symmetry acting only on the ψ fields, which is due to phase frustration from holes moving in a spin background. Some arguments to justify this from a microscopic point of view have been given in reference [4,14]. Consequently, this symmetry is associated with exotic statistics of the pertinent excitations [5], which is an exclusive feature of the planar spatial geometry. It should be noted that the existence of these gauge symmetries together with the appropriate constraints on single occupancy in antiferromagnetic materials, reduce the effective number of degrees of freedom to the physical degrees of freedom of the system. The ansatz (48) gives us a maximal symmetry G = SU(2) ⊗ U S (1).
The dynamics will be specified by a hamiltonian which will determine any spontaneous symmetry breaking that occurs. In the Hartree-Fock approximation we obtain the Hamiltonian [5]: where ∆ ij is a Hubbard-Stratonovich field. Using the gauge symmetries, then, of the (48) we can write where R ∈ SU(2) and U ∈ U S (1) are group elements 3 . These are phases of the above bilinears and, to a first (mean field) approximation can be considered as composites of the constituent fields, such as spinons z and holons ψ. Fluctuations about such ground states can then be considered by integrating over the constituent fields. Notice that the fermionic Hartree-Fock bilinear in (51) is a singlet under the abelian U S (1) group, as a result of the transformation of the holon fields, Ψ † i Ψ j → U ij Ψ † i Ψ j , as well as the transformations of t ij → U † ji t ij due to the spin frustration [4,14], and of the field ∆ ij → U † ji ∆ ij . This implies that in the effective action (50), the z-magnons are singlets under the U S (1) group, and couple only to the SU(2) part [16]. The resulting continuum action of this part is then a CP 1 σ-model coupled to the full SU(2) group. Details can be found in [16] and will not be repeated here.
As shown in [5], by integrating out the U S (1) factors, we obtain an effective action which is a functional of where i is a lattice site index of the microscopic theory, the constant κ depends on the microscopic parameters of the underlying theory, and M denotes a meson composite matrix which (in the bosonic theory of [5]) can be expanded in terms of the various meson composites: The notation for the various meson composites is as indicated in section 3. For the SU(2) group elements R (i)l we have the structure [5] Recalling that the gauge fields a (i)µ are actually composites of the constituent holons and spinons, it is natural to assume that the vector SU(2) gauge fields appearing in this way are identical to the corresponding vector meson fields A a µ (10) obtained after integration of the strongly-coupled U S (1) abelian group [5]. In this way, integration over constituent fields z, ψ (after inclusion of appropriate Jacobian factors [5]) can be transformed into an integration over meson fields, which express quantum fluctuations about the appropriate ground state.
From the analysis of sections 4 and 5 it becomes clear that the entire composite multiplet M (i) can be supersymmetrized. As we shall argue below, in this way, the dynamics of the composite system turns out to be equivalent to that of a N = 1 supersymmetric Abelian-Higgs model [11]. The fact that the ground state of dynamical supersymmetric systems yields zero energy by definition provides, then, strong support for the physical correctness of the identification of the local phase fluctuations of the Hartree-Fock ansatz with the vector composites (10).
Notice that, upon the identification of the gauge fields a µ with the vector composite fields A a µ , a = 1, 2, 3, we observe that there are Maxwell kinetic terms for the SU(2) gauge fields in the composite effective theory, and in this way the SU(2) interactions are promoted to fully dynamical ones. Such terms come by considering y (i)µ (52) for nearest neighbour lattice sites i, i +μ.
The Maxwell terms for the SU(2) gauge interactions will appear in the form of plaquette terms for the respective gauge link variables in the in the lattice action: where p denotes plaquettes, the β 2 are inverse couplings, β 2 ≡ β SU(2) ∝ 1/g 2 . and R p , are a product of the link variables over the plaquette p.
As argued in [8], for weakly coupled SU(2) interactions, the Grassmann fermionic variables χ a can be assembled into two Dirac spinors The inverse coupling β 2 must therefore be large on the other hand, in order to have a Dirac spinor representation of the holons [8]. In our case this coupling is related to the constant κ in (52). This is what will be assumed from now on. The analysis of [5] then, shows, that there is dynamical gauge symmetry breaking in the composite lagrnagian in the phase where a parity-invariant mass term appears, or equivalently in the phase where the scalar composite field Φ 3 (10) acquires a non-trivial vacuum expectation value (v.e.v.). < Φ 3 >= u = 0. Since u is a fermion (holon) condensate which is generated dynamically by means of the strongly coupled U S (1) interactions, whose coupling is formally infinite (of the order of the ultraviolet cut-off of the efective theory), quantum fluctuations of the condensate Φ 3 will be completely supressed. Two of the SU(2) gauge bosons, A 1,2 µ acquire masses in that case, proportional to the above vev, κ 2 u 2 → ∞, and, hence, they will decouple from the effective theory of light degrees of freedom, while the gauge field A 3 µ remains perturbatively massless. The symmetry breaking patterns of our theory, as well as the supersymmetry transformations derived in sections 4 and 5 imply [5,16] the following form of the effective composite action in the naive continuum limit: where χ is the real superpartner of the scalar singlet S, and φ = Φ 1 + iΦ 2 is a complex scalar, with ψ its Dirac (complex) spinor superpartner, constructed out of the two real superpartners of the scalar composites Φ i , i = 1, 2 of section 4. The Abelian gauge field A 3 µ , whose field strength is denoted by F µν , is the unbroken subgroup of the original SU(2) gauge group.
Notice that, despite the parity violating character of the composite excitations in the singlet supermultiplet, the corresponding terms in the action preserve parity 4 .
The action (57) is invariant under the following N = 1 supersymmetry: We observe that a major part of these transformations has already been derived in sections 4 and 5 by demanding a scalar N = 1 supersymmetry among the constituent spinon and holons. Unfortunately, however, at the level of bilinear composites, discussed in section 3, one cannot see the gauge-potential dependent parts in the corresponding gauge covariant derivatives in the transformation for δψ in (58). The latter part involves four fermions and hence it can only be derived from composite fields which contain higher order products of constituent fields in addition to the quadratic bilinear contributions studied here. As already mentioned, at a constituent level such coupling terms, which couple the vector and scalar N = 1 supermultiplets, appear to be irrelevant operators in a naive renormalization group sense. We hope to come to a more systematic study of such higher order terms in the composite operators in a future work.
Assuming for the purposes of this work that the N = 1 supersymmetric action of the composite fields has the form (57), one may make a comparison with the corresponding action appearing in [11]. In this case we observe that the action does not have potential terms, in contrast to [11], which implies that it constitutes a specific case of the actions of [11] corresponding to zero couplings e = λ = 0 in the notation of that work, where λ is a coupling constant in front of a Higgs-type potential for the doublets φ, and e is a coupling for interacting terms of the form ψΛφ − Λψφ * .
It is also important to notice that in our case with the trivial condition on the couplings e = λ = 0 the N = 1 supersymmetry can be elevated trivially to a N = 2 supersymmetry in our case. This is because the two real fermions Λ and χ can be assemblied in a single but complex Dirac fermion with the corresponding kinetic term in the effective lagrnagian: The action (57) then has an N = 2 supersymmetry invariance, as shown in [11]. The corresponding infinitesimal tranasformations are characterized by a complex parameter η ≡ εe iα , and they are equivalent to (58) with real parameter ε, followed by a phase transformation for the fermions Σ → e iα Σ and ψ → e iα ψ. In the more general case of [11], where e, λ = 0 one has that the N = 2 supersymmetry trnasformations imply a condition on the couplings e 2 = 8λ in the normalization of [11]. In our case, the absence of potential terms is compatible with the fact that the Abelian-Higgs model is obtained here from a spontaneous breakdown of the Georgi-Glashow model. Indeed in that model N = 2 supersymmetry implies a vanishing superpotential [13].
Notice that the analysis of [11], shows that the Noether supersymmetry currents corresponding to the model (57) are such that the pertinent supercharges satisfy a N = 2 algebra with central charge given by the topological charge of the Abelian-Higgs model. This is in accordance with the general arguments of [10], discussed in some detail in section 6. For completeness we will explicitly give the spinor charges and correspondingly for Q. The resulting centrally extended superalgebra is [11] Q α , Q β = 2 (γ 0 ) β α P 0 + δ β α T where P 0 = d 2 x 1 4 F 2 ij + 1 2 |D j (A 3 ) φ| 2 , with i, j = 1, 2, is the total energy and the central charge T is the topological charge of the model, as discussed in the previous section.
We also notice that the existence of an N = 2 dynamical supersymmetry at a composite level is compatible with the elevation of the N = 1 constituent supersymmetry of the CP 1 σ-model to a N = 2 supersymmetry, due to the existence of topological currents [10], as explained in section 6.

Discussion: N=2 Supersymmetry implies Superconductivity
The precise form of the effective theory is crucial for an understanding of the phase structure of the nodal liquid. As already mentioned, at a perturbative level, the gauge field A 3 µ is massless, and in fact the theory is superconducting [5]. However this may not be in general true when non-perturbative effects are taken into account, such as monopoles, which are instantons in the (2+1)-dimensional theory [17]. The monopole is a Euclidean configuration which behaves asymptotically aŝ where the caret indicates a unit vector and the tilde indicates the dual field tensor. In the N = 2 supersymmetric Georgi-Glashow model, or its N = 2 Abelian counterpart discussed here, the photon A 3 µ remains exactly massless, even at a non-perturbative level, due to the presence of Dirac (complex) fermions. This has been discussed in [13], and we shall not repeat the discussion here. Thus the nodal liquid at the supersymmetric point leads to the superconductivity mechanism proposed in [4]. It is worthy of noticing that such a masslessness is a property of the existence of complex fermions rather than composite supersymmetry. Simply, in our N = 2 supersymemtric case the existence of complex conposite fermions is a necessary consequence of the extended supersymmetry, and in this sense it is the constituent supersymmetry of spinon and holons rather than the composite one which guarantees superconductivity. Nevertheless, the existence of a N = 2 dynamical supersymmetric effective theory is important on its own in yielding exact non perturbative results on the phase structure in the way explained in [2].
The important point is that such a supersymmetry-argument based mechanism will work at strictly zero temperature, where supersymmetry is unbroken, and hence the above considerations may point towards a quantum critical (superconducting) point of the nodal liquid at the supersymmetric point of the parameter space of the microscopic model [8] 5 .
An interesting question concerns a transition from the superconducting to the pseudogap phase of the nodal liquid in our case, where indeed the supersymemtric point necessarily implies superconductivity of the nodal liquid of excitations. This may be provided by a simple variation of the doping concentration, which could take one away from the constituent supersymmetric point. In that case there is an explicit breaking of supersymmetry, and the masses of the spinon and holons are unequal. One may then arrive at the situation of [5], where the spinons are very massive and hence should be integrated out. In that case one is left with composites made only of holons, the light degrees of freedom in the problem. In such a case the effective theory is just the Georgi-Glashow model [16] without composite fermions (which may be thought of as being "very massive and thus decoupled" like the spinons). In such theories, according to the standard analysis [17], non-perturbative effects yield the photon A 3 µ a small mass, thereby leading to a pseudogap phase. If the above scenario is true, it would imply that there is a quantum critical point of the nodal liquid where the onset of superconductivity is identified with the onset of centrally extended N = 2 supersymmetries at both constituent and composite levels.
Much more work is needed before one arrives at a complete understanding of the underlying dynamics of the nodal liquids. In this work we have discussed a possible rôle of supersymmetry in yielding some exact results concerning the passage to a superconducting phase. In this respect we still lack a complete quantitiative derivation of the extended supersymmetric composite dynamics, given our present inabilitiy to handle analytically higher order terms of constituent field operators in the expressions for the compsoite fields. Such terms would ensure the extended N = 2 supersymmetries arising from the gauge coupling of the N = 1 composite vector and scalar supermultiplets. Neverhteless, the exciting features on the existence of extended supersymemtries, presented here, which could allow some exact results on the phase structure of the nodal liquids to be obtained, already open up interesting directions for theoretical research in such systems. We hope to come back to such studies in the near future.